The questions we address have a slightly different flavour: given formula \(f \in \mathfrak {T}\)  and variable \(x\)  occurring in \(f\) , we are interested in properties of the set of all values \(w\)  for which there is an assignment mapping \(x\)  to \(w\)  which makes the formula true. Thus, we consider the set of concrete values \(w\)  for which \(f\)  remains satisfiable once the variable \(x\)  has been replaced by \(w\) . Since we shall focus on theories in which variables represent words, we refer to the set of all such values \(w\)  as the language expressed by the variable \(x\)  in the formula \(f\) . In this respect, we extend the notion of languages expressible by word equations [1]} to arbitrary string-based logical theories. We say a language \(L\)  is expressed  by a formula if it contains a variable \(x\)  such that \(L\)  is the language expressed by \(x\)  in \(f\) . We say that \(L\)  is expressible in a theory \(\mathfrak {T}\)  if there exists a formula \(f \in \mathfrak {T}\)  and variable \(x\)  occurring in \(f\)  such that \(L\)  is expressed by \(x\)  in \(f\) .
